On groups acting freely on a tree

Abstract.

Suppose we are given a group \(\mit\Gamma \) and a tree X on which \(\mit\Gamma \) acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers \(a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \}\) if \(d\rightarrow \infty \), where v, vo are some fixed vertices in X.¶ In this paper we consider the case where \(\mit\Gamma \) is a finitely generated group acting freely on a tree X. The growth function \(\textstyle\sum\limits a_d x^d\) is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where \(\mit\Gamma \) is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.